Exponential Functions
A free Algebra II lesson from the “Exponential Models” unit, with a worked example and practice problems including step-by-step solutions.
An exponential function has a variable in the exponent, such as y = a(b)^x. The base b is the multiplier from one step to the next. If b is greater than 1, the function shows growth; if b is between 0 and 1, it shows decay. Exponential functions matter because they model repeated percent change, compound interest, population growth, and depreciation. When practicing, identify the starting value and the multiplier before calculating. A common mistake is adding the same amount each step; exponential patterns multiply by the same factor instead.
What you'll learn
- Evaluate exponential functions
- Distinguish growth and decay
- Use percent change models
Worked example
Problem. Evaluate f(x) = 3(2^x) when x = 4.
- Substitute 4 for x.
- 2^4 = 16.
- 3 x 16 = 48.
Answer: 48
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Evaluate f(x) = 2^x when x = 5.
Show solution
- 2^5 = 32.
Answer: 32
2. Evaluate g(x) = 5(2^x) when x = 3.
Show solution
- 2^3 = 8.
- 5 x 8 = 40.
Answer: 40
3. y = 7(1.2)^x represents...
Choices: Growth · Decay · Linear change · No change
Show solution
- The base 1.2 is greater than 1.
Answer: Growth
4. y = 100(0.8)^x represents...
Choices: Decay · Growth · Quadratic change · No function
Show solution
- The base is between 0 and 1.
Answer: Decay
5. A value starts at 200 and grows by 10%. What multiplier is used each period?
Show solution
- Growth multiplier is 1 + 0.10.
Answer: 1.1
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